The book is the second revised edition of the textbook on Real Analysis of functions of one variable (first published in 2001 (Zbl 1093.26003)). The main feature of the book is its instructive character. The author tries to explain the most known questions of calculus and analysis on very simple examples and counterexamples by using geometrical illustration, analogies in other mathematical disciplines. In particular, he uses the ideas of recent books which are brilliant additions to a standard course of Analysis [{\it J. Appell}, Analysis in examples and counterexamples. An introduction to the theory of real functions. Springer-Lehrbuch. Berlin: Springer (2009; Zbl 1168.26001)]; [{\it V. M. Shibinskii}, Examples and counter-examples in a course on mathematical analysis. Moskva: Vysshaya Shkola (2007; Zbl 1198.26001)]. Another interesting characteristic of the book is the utilization of elements of foundation of the Set Theory, Number Theory and Analysis. This helps the reader to reach a better understanding of the main ideas of Real Analysis. In this sense the book is self-contained and can be recommended to private study or to the study of analysis on a higher level. The structure of the book is the following. It starts with some preliminaries presented in the Chapter 0 “Sets, Relations and Mappings”. This is the basis for a presentation of the foundations of Analysis and Numbers Systems (Chapter 1 “Foundation of Analysis”, Chapter 2 “Systems of Real Numbers”). In the main part of the book only general ideas are given. A more formal presentation is contained in Appendix A “Systems of Sets, Relations and Partitions” and in Appendix B “Construction of Real Numbers”. Standard results from the Agenda of Analysis 1 for first year students in Pure and Applied Mathematics are presented in the next chapters. In a sense this presentation is close to the traditional classical books of Analysis in Germany, France and Russia. This material is gathered in Chapter 3 “Infinite Series”, Chapter 4 “Continuous Functions in one Variable”, Chapter 5 “Differential Calculus in one Variable”, Chapter 6 “Elementary Transcendental Functions”, Chapter 7 “Integral Calculus”, Chapter 8 “Riemann Integral”. Additionally, the author describes certain elementary facts of Complex Analysis in Appendix C “Elementary Complex Analysis”. Surely, the book can be recommended as a textbook on Analysis 1 for first year students in Pure and Applied Mathematics.
Reviewer: Sergei V. Rogosin (Minsk)